Why do they persist to call this Tai Ji Push hands instead of just calling it Shuai Jiao? With the exception of the starting position to me it looks like Shuai Jiao with out the jacket. I am by no means an authority on Tai ji push hands so is there something I am not seeing here?

As far as I can tell, push hands IS (or rather, should be) Shuai Jiao without a jacket. Sort of the Greco-Roman to Shuai Jiao's Judo.

The fool thinks himself immortal,
If he hold back from battle;
But old age will grant him no truce,
Even if spears spare him.

So what is the ruleset and scoring system videos 1 & 3 are competing under? Is there one unified approach to push hands competitions?

There isn't even a unified approach to push hands competitions within the same tournament. There a zillion versions and a few major rulesets.

#1 almost certainly doesn't allow any contact to the head, back of the neck, direct grabs of the leg, or likely even both hands on the back at the same time for long. #3 likely allows all those things.

Scores are generally something on the order of: 1 point (or 2) for making one's opponent hit the floor with anything other than their feet, less if you fall down to do it too. It mght be first to 3, or first to 5, or most points within some period of time as well.

There isn't even a unified approach to push hands competitions within the same tournament. There a zillion versions and a few major rulesets.

Which is what makes most tournament play so frustrating. Also they generally don't post the rules before the tournament, and sometimes each of the judges seems to be looking for different things. Also in the US the tournaments are usually micromanaged by four (four!) judges who stand on top of you stopping play every 5 seconds because you "used force." And a lot of the rules are just fricking stupid (no touching the opponent's back, no touching the opponent's neck, no touching one of his arms with two of yours, yada yada yada).

I actually thought that the guy in the black uniform with the red sash at the end of the first vid was best. He was relaxed and rooted and his pushes were controlled and relatively precise. The rest of the stuff just looked like sloppy standup (and falldown) grappling.

I think that push hands should be different from grappling. For instance, in "Mastering Jujitsu," Renzo writes: "Rather than confront strength with strength, jujitsu confronts strength with efficient technique and strategy so that the weaker fighter can attain victory. In practice, this means applying a high percentage of your total strength to a low percentage of your opponent's strength."

So in grappling (as in all types of sparring), whether in CMA or BJJ,we are trying to locate the opponent's weakness and apply a large percentage of our body's force to it to defeat him (or her). But in push hands (not shuai jiao), I've always felt that we're probing the partner's structure to help him (or her) build better root and posture, and the ability to neutralize and redirect incoming force. The focus is really on cooperation not competition. Kauz writes in his book "Push Hands": "I believe we err in attempting to make tai chi more sportive. Tai chi offers us an opportunity to relax, to open up, to ground and center ourselves...Students who are attracted to competitive push hands will develop some level of skill in avoiding and resisting pushes, and in pushing, but they will fail woefully in achieving the much more important and valuable changes in themselves push hands could make possible...Push hands done according to tai chi principles is alien to us, to the way we conduct ourselves in daily life. That is, invariably if we are pushed we resist. If our own attempt to push meets resistance, we push harder."

So at this point in my life I've concluded that students make more progress in tai chi when push hands is viewed as a noncompetitive, but "alive," drill. Grappling and sparring are also necessary, but they are different tools for progress, with different aims, that should be viewed differently. As for tournaments like the ones shown above, maybe they should be called "T'ai Chi Grappling" as opposed to "Push Hands."

Which is what makes most tournament play so frustrating. Also they generally don't post the rules before the tournament, and sometimes each of the judges seems to be looking for different things. Also in the US the tournaments are usually micromanaged by four (four!) judges who stand on top of you stopping play every 5 seconds because you "used force." And a lot of the rules are just fricking stupid (no touching the opponent's back, no touching the opponent's neck, no touching one of his arms with two of yours, yada yada yada).